Hebbian Learning

Next Assignment ,[object Object],[object Object],[object Object],[object Object]

ART1 Demo Increasing vigilance causes the network to be more selective, to introduce a new prototype when the fit is not good. Try different patterns

Hebb’s Postulate “ When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.” D. O. Hebb, 1949 A B In other words, when a weight contributes to firing a neuron, the weight is increased. (If the neuron doesn’t fire, then it is not).

Colloquial Corollaries ,[object Object]

Colloquial Corollaries? ,[object Object]

Generalized Hebb Rule ,[object Object],[object Object]

Flavors of Hebbian Learning ,[object Object],[object Object],[object Object],[object Object]

Unsupervised Hebbian Learning (aka Associative Learning)

Simple Associative Network input output

Banana Associator Unconditioned Stimulus Conditioned Stimulus Didn’t Pavlov anticipate this?

Banana Associator Demo can be toggled

Unsupervised Hebb Rule Vector Form: Training Sequence: actual response input

Learning Banana Smell Initial Weights: Training Sequence: First Iteration (sight fails, smell present):  = 1 unconditioned (shape) conditioned (smell) a 1   h a r d l i m w 0 p 0 1   w 0   p 1   0.5 – +   h a r d l i m 1 0  0 1  0.5 – +   0 (no banana) = = =

Example Second Iteration (sight works, smell present): Third Iteration (sight fails, smell present): Banana will now be detected if either sensor works. a 2   h a r d l i m w 0 p 0 2   w 1   p 2   0.5 – +   h a r d l i m 1 1  0 1  0.5 – +   1 (banana) = = =

Problems with Hebb Rule ,[object Object],[object Object]

Hebb Rule with Decay This keeps the weight matrix from growing without bound, which can be demonstrated by setting both a i and p j to 1:

Example: Banana Associator First Iteration (sight fails, smell present): Second Iteration (sight works, smell present):  = 0.1  = 1 a 1   h a r d l i m w 0 p 0 1   w 0   p 1   0.5 – +   h a r d l i m 1 0  0 1  0.5 – +   0 (no banana) = = = a 2   h a r d l i m w 0 p 0 2   w 1   p 2   0.5 – +   h a r d l i m 1 1  0 1  0.5 – +   1 (banana) = = =

Example Third Iteration (sight fails, smell present):

General Decay Demo no decay larger decay w i j m a x   - - - =

Problem of Hebb with Decay Associations will be lost if stimuli are not occasionally presented. If a i = 0, then If  = 0, this becomes Therefore the weight decays by 10% at each iteration where there is no stimulus.

Solution to Hebb Decay Problem ,[object Object],[object Object]

Instar Operation The instar will be active when or For normalized vectors, the largest inner product occurs when the angle between the weight vector and the input vector is zero -- the input vector is equal to the weight vector. The rows of a weight matrix represent patterns to be recognized. w T 1 p w 1 p  cos b –  =

Vector Recognition If we set the instar will only be active when   =  0. If we set the instar will be active for a range of angles. As b is increased, the more patterns there will be (over a wider range of  ) which will activate the instar. b w 1 p – = b w 1 p – > w 1

Instar Rule Hebb with Decay Modify so that learning and forgetting will only occur when the neuron is active - Instar Rule: or Vector Form: w i j q   w i j q 1 –    a i q   p j q    a i q   w q 1 –   – + = i j

Graphical Representation For the case where the instar is active ( a i = 1): or For the case where the instar is inactive ( a i = 0):

Instar Demo weight vector input vector  W

Outstar Operation Suppose we want the outstar to recall a certain pattern a * whenever the input p = 1 is presented to the network. Let Then, when p = 1 and the pattern is correctly recalled. The columns of a weight matrix represent patterns to be recalled.

Outstar Rule For the instar rule we made the weight decay term of the Hebb rule proportional to the output of the network. For the outstar rule we make the weight decay term proportional to the input of the network. If we make the decay rate  equal to the learning rate  , Vector Form:

Linear Associator Training Set:

Hebb Rule Presynaptic Signal Postsynaptic Signal Simplified Form: Supervised Form: Matrix Form: actual output input pattern desired output

Batch Operation Matrix Form: (Zero Initial Weights)  W t 1 t 2  t Q p 1 T p 2 T p Q T T P T = = T t 1 t 2  t Q = P p 1 p 2  p Q =

Performance Analysis Case I, input patterns are orthogonal. Therefore the network output equals the target: Case II, input patterns are normalized, but not orthogonal. Error term 0 q k  =

Example Banana Apple Normalized Prototype Patterns Weight Matrix (Hebb Rule): Tests: Banana Apple

Pseudoinverse Rule - (1) Performance Index: Matrix Form: Mean-squared error T t 1 t 2  t Q = P p 1 p 2  p Q = || E || 2 e i j 2 j  i  =

Pseudoinverse Rule - (2) Minimize: If an inverse exists for P , F ( W ) can be made zero: When an inverse does not exist F ( W ) can be minimized using the pseudoinverse:

Relationship to the Hebb Rule Hebb Rule Pseudoinverse Rule If the prototype patterns are orthonormal: W T P T =

Tests 50% Occluded 67% Occluded Noisy Patterns (7 pixels)

Spectrum of Hebbian Learning Basic Supervised Rule: Supervised with Learning Rate: Smoothing: Delta Rule: Unsupervised: target actual

Hebbian Learning

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Hebbian Learning